We give an original representation integral formula for the resolvent families associated to the fractional Cauchy equation with memory effects

$${}_C{D}_t^{\alpha }u(t)-Au(t)+{\int }_0^t\beta (t-s)Au(s)ds=f(t,u(t)),t\in [0,T],T>0,$$

where $u(0)=u_0\in X$ and A is a sectorial operator on a Banach space X. Moreover, we get spatial bounds for the resolvent families in order to study global or blow up mild solutions when the nonlinearity f satisfies a locally Lipschitz condition. The case of critical nonlinearities is also treated.

中文翻译：

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具有记忆效应的分数柯西问题

我们给出了与具有记忆效应的分数柯西方程相关的分解物族的原始表示积分公式

$${}_C{d}_{Ť}^{\alpha }ü（Ť）-\mathrm{一种}ü（Ť）+{\int }_0^{Ť}\beta （Ť-s）\mathrm{一种}ü（s）ds=F（Ť，ü（Ť）），Ť\in [0，Ť]，Ť>0，$$

哪里 $ü（0）={ü}_0\in X$并且A是Banach空间X上的部门算子。此外，当非线性f满足局部Lipschitz条件时，为了研究全局变量或分解温和解，我们获得了分解族的空间边界。还处理了临界非线性的情况。